On the Integrate-and-Fire model with superlinear drift

Benoît Perthame; Clément Rieutord; Delphine Salort

Home/Archive/arXiv

arXiv · 2610.41497

On the Integrate-and-Fire model with superlinear drift

Benoît Perthame·Clément Rieutord·Delphine Salort

View original source ↗Read source document ↗Export metadata ↓

Abstract

The Nonlinear Integrate-and-Fire equation is a fundamental model for neural assemblies, describing the dynamics of the density of neurons with a given potential. Its standard formulation uses a given firing potential, and is known to undergo finite-time blow-up for excitatory connections. To fit experimental observations, it has been advocated to use superlinear drifts. We show how, in this situation, it makes sense to set the equation on the whole line. The main question we address is whether in these circumstances the solution still exhibits finite-time blow-up. We study the construction of solutions by a fixed point and, using the framework of \cite{carrilloQualitativePropertiesSolutions2015}, we develop novel a priori estimates which use specifically the superlinearity of the drift at $+\infty$. In particular, we obtain Ces{à}ro-mean bounds on the activity, as well as exponential convergence towards the unique steady state in the weakly connected regime. Several critical steps rely on controls depending solely on the $L^1$ norm of the solution or coefficients.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Benoît Perthame, Clément Rieutord, Delphine Salort. 2026-10-30. On the Integrate-and-Fire model with superlinear drift. https://arxiv.org/abs/2610.41497

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Sharp Poincaré inequalities for the logarithmic smoothness of the Gauss map on hypersurfaces

We establish sharp lower bounds for two geometric functionals which evaluate the global variation of the Gauss map on the boundary of bounded open sets $Ω\subset\mathbb{R}^d$. These functionals arise as the endpoint case $s=0$ of the Gagliardo seminorm in $W^{s,p}(\partialΩ)$ of the normal vector field, for $p\in\{1,2\}$. We also show that balls are the unique minimizers. In addition, some applications to the fractional setting $s>0$ are given.

math.AP↗

arXiv2026

Upper estimates of the lifespan for the fractional wave equations with time-dependent damping and a power nonlinearity of subcritical and critical Fujita exponent

In this paper, we investigate the Cauchy problem of the fractional wave equation with time-dependent damping and the source nonlinearity $f(u)\approx |u|^p$: $$ \begin{cases} \partial_t^2u(t,x)+(-Δ)^{σ/2} u(t,x)+b(t) \partial_t u(t,x) =f(u(t,x)),\ &(t,x)\ \in [0,T)\times \mathbb{R}^N,\\ u(0,x)=u_0(x),\ \partial_tu(0,x)=u_1(x),\ &x\ \in\ \mathbb{R}^N, \end{cases} $$ where the damping coefficient behaves as $b(t)\approx (1+t)^{-β}$. We first establish local existence and uniqueness of mild and strong solutions in weighted Sobolev spaces by deriving weighted Sobolev estimates for the linear propagators and overcoming the spatial-weight coupling in the nonlinearity through the Bessel potential estimates. Then, in the subcritical and critical cases $1<p\leq p_c:=1+\frac σN$, we derive the upper estimates of the lifespan for the fractional Laplacian with $0<σ<2$ and time-dependent damping $β\in [-1, 1)$ via the framework of ordinary differential inequalities. The blow-up results, together with the global existence in the supercritical case obtained in \cite{LI}, show that the critical exponent for the fractional wave equation is $p_c=1+\fracσ{N}$ for $0<σ<2$. Moreover, combining with the lower estimate of lifespan derived in \cite{LI}, we conclude that the estimate in this paper is sharp. Note that our result for the critical case is completely new even in the classical case $b(t)=1$. We also consider the case of $β=1$, and obtain the upper estimate of the lifespan.

math.AP↗

arXiv2026

On the stationary Navier-Stokes equations in distorted pipes under energy-stable outflow boundary conditions

The steady motion of a viscous incompressible fluid in distorted pipes, of finite length, is modeled through the Navier-Stokes equations with mixed boundary conditions: the inflow is given by an arbitrary member of the Lions-Magenes class with positive influx, and the fluid motion is subject to a directional do-nothing boundary condition on the outlet, together with the standard no-slip assumption on the remaining walls of the domain. Existence of a weak solution to such Navier-Stokes system is proved without any restriction on the data (that is, inlet velocity and external force) by means of the Leray-Schauder Principle, in which the required a priori estimate is obtained by a contradiction argument that employs Bernoulli's law for solutions of the stationary Euler equations, as well as some properties of harmonic divergence-free vector fields. Under a suitable smallness assumption on the data, we also prove the unique solvability of the boundary-value problem.

math.AP↗